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I'm reading Sipser's Introduction to the Theory of Computation (3rd edition). In chapter 0 (pg. 2), he says we don't know the answer to "what makes some problems computationally hard and others easy," however, he then states that "researchers have discovered an elegant scheme for classifying problems according to their computational difficulty. Using this scheme, we can demonstrate a method for giving evidence that certain problems are computationally hard, even if we are unable to prove that they are."

So my question is: HOW is it possible to classify problems according to their computational difficulty, if we don't even know what makes a problem computationally easy/hard in the first place?

Also, what/where is this "scheme" that does this classifying. (I did some googling and couldn't find anything)

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That's what you get when you distill a whole bunch of theory to a wider audience.

In his book, Sipser addresses a general audience at the undergraduate level, possibly with no notion of computability theory; hence, he can only hint at concepts which are to be given a more formal treatment later on in the book. The part you cite is from chapter 0 (i.e., not really a chapter), whereas the material for complexity theory only appears at the end (i.e., part three). This is why the passage is so fuzzy. Most likely it is intended only as motivation and to give a broad overview for the topics to be covered in the book.

The "scheme" Sipser is talking about are reductions. If we know a problem $A$ is reducible to a problem $B$, then we know $B$ is at least as hard as $A$. (Incidentally, this is also why it is common practice to denote reductions with a "$\le$" sign.) This gives us a way of ordering problems according to their difficulty, at least for those having reductions we are aware of. As Sipser states, though, by using only reductions "we are unable to prove" whether the problems are really hard or not; reductions only give us relative, not absolute notions of hardness. This is why separation results are still rare in modern complexity theory: We have a bunch of reduction (e.g., NP-completeness) results, but only a handful of separation results (e.g., the time and space hierarchy theorems).

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  • $\begingroup$ I appreciate the thorough answer. Vincenzo (one of the commentors) mentioned that Sipser discusses this in Ch 5 & 7, which I'll hopefully get to eventually! $\endgroup$ – Johan von Adden Mar 29 at 10:12
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HOW is it possible to classify problems according to their computational difficulty, if we don't even know what makes a problem computationally easy/hard in the first place?

I think the point that the piece is trying to make is that we know how to determine whether individual problems are easy or hard, even though we don't have an over-arching theory of why the hard ones are hard and the easy ones are easy. Just like you can classify people according to their weight, even though you don't know why they have the weight they do.

I should emphasise that in most cases, "hard" means "seem to be hard". You've probably heard of NP-complete problems. We don't know for certain that these problems have no efficient algorithm (by the standard definition of "efficient") but nobody has been able find an efficient algorithm for any of them in nearly 50 years of trying, and finding an efficient algorithm for just one of them would give efficient algorithms for all of them.

Also, what/where is this "scheme"

Complexity classes, the relationships between them, and the concept of reductions for transforming one problem into another.

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The "scheme" is based on the ideas of reductions among problems and completeness of problems, which are described in Chapters 5 and 7 of Sipser's book.

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